nLab IPC-property

Idea

The IPC-property (“inductive limit product commutation property”) is a technical condition on a category $A$ which ensures that presheaves with values in $A$ have a good notion of sheafification.

Definition

A category $A$ is said to satisfy the IPC-property if

• it admits small products and filtered colimits;

• for

  • every indexed set $\{I_s\}_{s \in S}$ of small filtered categories and

  • for any indexed set $\{\alpha_s \colon I_s \to A\}$ of functors

  • indexed by a small set $S$

  the canonical morphism

  $$colim \left( \prod_s I_s \xrightarrow { k \mapsto \prod_s \alpha_s(\pi_s(k)) } A \right) \longrightarrow \prod_s colim \alpha_s$$

  is an isomorphism.

References

• Masaki Kashiwara, Pierre Schapira; Def. 3.1.10 in in: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

It is invoked for sheafification in section 17.4 there.